Question

Josh Wilson is traveling to California from South Carolina for summer vacation. The states are approximately 2500 miles apart driving cross-country. He stops at his parent's house in Oldham County, Texas, which is (12x 106) miles away from South Carolina and (5x + 685) miles away from California. Determine if Josh's parent's house in Texas the midpoint between South Carolina and California? Justify your answer.

Answer 1

Answer: Yes, Josh's parent's house in Texas is the midpoint between South Carolina and California.

Solution:

Distance from South Carolina to California: d=2,500 miles

Distance from South Carolina to Josh's parent's house in Texas: d1=12x-106

Distance from Josh's parent's house in Texas to California: d2=5x+685

Josh's parent's house in Texas is the midpoint between South Carolina and California if:

d1=d2=d/2→d1=d2=2,500 miles / 2→d1=d2=1,250 miles

Now, we know:

d1+d2=d

Replacing d1 by 12x-106; d2 by 5x+685; and d by 2,500:

(12x-106)+(5x+685)=2,500

12x-106+5x+685=2,500

Adding similar terms:

17x+579=2,500

Solving for x: Sutracting 579 both sides of the equation:

17x+579-579=2,500-579

17x=1,921

Dividing both sides of the equation by 17:

17x/17=1,921/17

x=113

With x we can calculate d1 and d2

d1=12x-106→d1=12(113)-106→d1=1,356-106→d1=1,250

d2=5x+685→d2=5(113)+685→d2=565+685→d2=1,250

Like d1=d2=1,250, Josh's parent's house in Texas is the midpoint between South Carolina and California